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2008•Bulletin of the Belgian Mathematical Society - Simon StevinOpen access

Invariant Subspaces Of Toeplitz Operators And Uniform Algebras

Takahiko Nakazi

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Abstract

Let $T_\phi $ be a Toeplitz operator on the one variable Hardy space $H^2$. We show that if $T_\phi $ has a nontrivial invariant subspace in the set of invariant subspaces of $T_z$ then $\phi $ belongs to $H^\infty $. In fact, we also study such a problem for the several variables Hardy space $H^2$.

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Let $T_\phi $ be a Toeplitz operator on the one variable Hardy space $H^2$. We show that if $T_\phi $ has a nontrivial invariant subspace in the set of invariant subspaces of $T_z$ then $\phi $ belongs to $H^\infty $. In fact, we also study such a problem for the several variables Hardy space $H^2$.

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Available abstract

Let $T_\phi $ be a Toeplitz operator on the one variable Hardy space $H^2$. We show that if $T_\phi $ has a nontrivial invariant subspace in the set of invariant subspaces of $T_z$ then $\phi $ belongs to $H^\infty $. In fact, we also study such a problem for the several variables Hardy space $H^2$.

Key concepts: Linear subspace, Toeplitz matrix, Reflexive operator algebra, Hardy space, Mathematics, Invariant (physics), Invariant subspace, Pure mathematics

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